Zygmund defined a class of kernels that generalized the hilbert transform and riesz transform at the same time to higher dimensional settings and they studied the lp boundedness of these kernels. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. Commutators of singular integral operators satisfying a. Questions of this type have been traditionally answered using the theory of singular integrals and using harmonic analysis. These results are new even in the euclidean space and are the. Jul 04, 2007 pdf file 4047 kb article info and citation. We develop a wide general theory of bilinear biparameter singular integrals t. A systematic treatment of multilinear calderonzygmund operators is presented. In the 1980s and early 1990s, coifman, weiss, and christ noticed that the theory of calderonzygmund operators can be generalized from euclidean spaces to spaces of homogeneous type. In the third of the distinguished lecture series given by eli stein here at ucla, eli presented a slightly different topic, which is work in preparation with alex nagel, fulvio ricci, and steve wainger, on algebras of singular integral operators which are sensitive to multiple different geometries in a nilpotent lie group. Instead, an endpoint theory was provided for this operator, such as the weak type llog. Pseudolocalization of singular integrals and noncommutative. The big contribution to the general theory of operators that share similar properties with the hilbert transform is due to the work of.
Finally, some applications of the main results are demonstrated by examples. Giuseppe mingione the standard calderonzygmund theory deals with optimal integrability properties of. The zygmundcalderon lectures are named after antoni zygmund 19001992 and alberto calderon 19201998. Motivation and description of the main results the calderonzygmund theory of singular integral operators was carried over to the vectorvalued setting by several authors, but we have been especially influenced in writing this paper by the work of benedek, calderon, and panzone 2. Our assumptions on the operators are minimal and our result applies to an array of. The main result of this paper is the following weak type llog. This volume of essays on fourier analysis and its applications arose from a conference held at princeton university in may 1991 in honour of the mathematician elias steins 60th birthday. An example in the theory of singular integrals, studia math. Pdf in this paper, we study some weighted norm inequalities related to the calderonzygmund singular integrals. A course in calderonzygmund singular integrals and applications.
Rd c, where rd denotes euclidean space and c denotes. A crash course in calderonzygmund singular integrals and. The theory of singular integrals started with the study of singular. I have written some notes on littlewoodpaley theory for a masterclass, which i thought i would share here as well. A geometric approach to the calderonzygmund estimates 3 corollary 1. The canonical example of such an operator is the hilbert transform, which is given by. Singular integrals and differentiability properties of. In mathematics, the calderonzygmund lemma is a fundamental result in fourier analysis, harmonic analysis, and singular integrals. In order to state the result we must first define some terms.
For any 0 pseudodifferential operators and the cauchy integral of calderon lecture notes in mathematics on free shipping on qualified orders. In this talk, i would like to report several results on sobolev regularity theory of calderon zygmund type for weak solutions of elliptic equations in divergence form with degenerate, singular coefficients. Huixia mo, hongyang xue submitted on 10 jun 2015 v1, last revised 16 nov 2015 this version, v2. This is one of the first general theories of multiparameter singular integrals that goes beyond the product theory of singular integrals and their analogs. Some better than expected estimates for certain multilinear operators are presented too.
Extremal problems in interpolation theory, whitney. Calderonzygmund theory for secondorder elliptic systems on. Boundedness of commutators generated by mth calderonzygmund type singular integrals and local campanato functions on generalized local morrey spaces authors. Broadly speaking a singular integral is an integral operator. Lecture notes on singular integrals, projections, multipliers. The calderonzygmund theory for elliptic problems with. This book introduces some important progress in the theory of calderonzygmund singular integrals, oscillatory singular integrals, and littlewoodpaley theory over the last decade. Coifman algebras of singular integral operators 18 a. Lecture notes on singular integrals, projections, multipliers and. Pdf bilinear calderonzygmund theory on product spaces. H is a limiting case just outside the scope of that theory, for.
Calderonzygmund theory, multilinear operators, pseudodifferential op erators, interpolation, t1 theorem, littlewoodpaley theory, wavelets, schurs test. Sobolev spaces, distributions, interpolation inequalities 7. Stein 1 n otices of the ams v olume 45, number 9 t he subject matter of this essay is alberto calderons pivotal role in the creation of the modern theory of singular integrals. Many important properties of singular integrals have been thoroughly studied for calderonzygmund operators. Calderon zygmund regularity estimates for weak solutions of singular and degenerate elliptic equations. Calderon had the good fortune of work ing with antoni zygmund, who was at first his. Commutators of multilinear singular integral operators. The theory of calder onzygmund singular integrals has expanded and interacted with many areas of mathematics and, as several articles in these proceed ings show, it continues to be a very strong and active area of research. In this dissertation we further develop the theory of commutators of multilinear singular integral operators with pointwise multiplication. Revista matematica iberoamericana, issn 022230, vol. There are noperators called the riesz transforms with similar properties.
First, we prove a dyadic representation theorem starting from t1assumptions and apply it to show many estimates. The seminal work on singular integrals by calder on and zygmund, as originated in 5, and the real variable methods later developed have played a crucial and in uential role in modern harmonic analysis. To that aim, we have developed a pseudolocalization principle for singular integrals which is of independent interest, even in the classical theory. Zygmund was on the faculty of the university of chicago from 1947 until his retirement in 1980. A crash course in calderonzygmund singular integrals and applications joaquim bruna february 4, 2016 part i. Multilinear calderonzygmund singular integrals 3 further use of interpolation provides an improved version of 4 in which lp. This work presents full proofs of vitushkins conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. It includes some important research results by the authors and their co.
Introduction in the last ten years, it has been of great interest to obtain optimal operator. Bilinear littlewoodpaley square functions and singular. Calderonzygmund capacities and operators on nonhomogeneous. These results have frequently been cited in many published papers. New maximal functions and multiple weights for the multilinear calderonzygmund theory andrei k. The theory has had quite a success in the solution of many problems in both real. Pdf convolution calderonzygmund singular integral operators. References i have used in preparing these notes include steins singular integrals and differentiability properties of. The theory developed includes strong type and endpoint weak type estimates, interpolation, a multilinear t1 theorem, and a variety of results regarding multilinear multiplier operators. A third type of integrals suggested by the theory of spherical summa. Buy singular integrals and differentiability properties of functions. Boundedness of maximal singular integral operators on spaces of homogeneous type and its applications hu, guoen, yang, dachun, and yang, dongyong, journal of the mathematical society of japan, 2007. It is named for the mathematicians alberto calderon and antoni zygmund.
This is the first part, covering some motivation, the case of a single frequency projection and its vectorvalued generalisation. Towards a nonlinear calderon zygmund theory giuseppe mingione to tadeusz iwaniec on his 60th birthday contents 1. A multisublinear maximal operator that acts on the product of m lebesgue spaces and is smaller that the mfold product of the hardylittlewood. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Boundedness of calderonzygmund singular integral operators. Cordes boundary value problems for secondorder parabolic equations 82 e. Several tools and techniques for the study of such operators are discussed.
Stefanov, lp bounds for singular integrals and maximal singular integrals with rough kernels, indiana univ. Kx,yfydy, whose associated kernel function k is singular along the diagonal x y. While the new method is presented in great detail in the second part, the first is mainly devoted to the prerequisites needed for a selfcontained. A geometric approach to the calderonzygmund estimates abstract. The hilbert transform on the real line l is a typical example. Singular integrals 1 calderonzygmund decomposition people. In many cases the minimizers constructed in the book are stable i. L estimate for multilinear commutator of the singular integral operator defined in theorem 3. Fractional derivatives and pseudo differential operators 8. In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Contents 1 a brief introduction to the a p theory of weights 4 2 lp properties of functions with controlled oscillation 9. Weak estimates, marcinkiewicz interpolation theorem lorentz spaces 3. Arc spaces, motivic integration and stringy invariants veys, willem, 2006.
Singular integrals nowadays appear in partial di erential equations, several complex variables, operator theory, and other areas of analysis. Bilinear integral operators appear naturally in many contexts and have been a source of many challenging problems in harmonic analysis. Pdf a survey of known results in the theory of convolution type. The tb theorem provides sufficient conditions for a singular integral operator to be a calderonzygmund operator, that is for a singular integral operator associated to a calderonzygmund kernel to be bounded on l 2.
Interpolation of operators riesz theorem on interpolation in lp spaces. Moreover, in the following, when treating equations as in 1. In that great enterprise calderon had the good fortune of working with antoni zygmund, who was. Calderonzygmund estimates for parabolic measure data equations. Moreover, the theory of the model operators themselves and the us. In their pioneering papers which appeared in american journal of mathematics1956 a. Fourier integrals, plancheral, parceval identities. Calder onzygmund theory of singular integrals handout. Calderonzygmund capacities and operators on nonhomogeneous spaces cbms regional conference series in mathematics 9780821832523. On weighted hadamardtype singular integrals and their applications xu, yong jia, abstract and applied analysis, 2007. We dominate nonintegral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Chapter xi calderon zygmund singular integral operators 1. Calderonzygmund regularity estimates for weak solutions of. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the cauchy transforms and other singular integrals.
This created the chicago school of hard analysis sometimes simply. This article deals with the boundedness properties of singular integrals that generalize the double hilbert transform on product domains. For every 1 a p a 8 one can extend any singular integral operator t to an operator bounded from lp n lp. Introduction to calderonzygmund operators in this lecture we will start our study of calder onzygmund operators in the oneparameter setting. Variants of the calderonzygmund theory for lpspaces. In this paper we obtain the weak type 1,1 boundedness of calderonzygmund operators acting over operatorvalued functions. Zygmundcalderon lectures in analysis department of. Full text full text is available as a scanned copy of the original print version.
L estimate and the weak type h 1, l 1 estimate see 20, 24. Pseudolocalization of singular integrals and noncommutative calderonzygmund theory. Zygmund remarks about theory of singular integral operators 74 h. There are many other textbooks on measure theory that cover most or all of the material in the present book, as well as much more, perhaps from somewhat different view points. Generally speaking, the commutator of two operators is itself an operator that measures the changes which occur when switching the order in which the commuted operators are being applied. Calderonzygmund theory of singular integrals handout. A variety of results regarding multilinear calderonzygmund sin gular integral operators is systematically presented. Some aspects of vectorvalued singular integrals citeseerx. I am going to survey a number of recent nonlinear regularity results that, put together, outline what might be considered a nonlinear version of caldero. Gilbargtrudinger 5, geometric measure theory, geometric group theory, ergodic theory and dynamical systems, and differential topology and geometry. New maximal functions and multiple weights for the.
1415 922 77 173 1033 1389 1336 1018 404 146 582 637 958 1313 1066 468 945 692 568 901 678 355 409 804 769 1126 302 276 1437 397 770 1286 155 370 807 1425 1435 84 669 1424 1271 645